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Algorithmic problems on convex sets
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Algorithmic problems on convex sets
Many problems in mathematical programming can be formulated as problems on convex sets or convex bodies. Six kinds of problems are particularly important: optimization, violation, validity, separation, membership and emptiness. Each of these problems has a strong (exact) variant, and a weak (approximate) variant.
In all problem descriptions, K denotes a compact and convex set in Rn.
The strong variants of the problems are:
Closely related to the problems on convex sets is the following problem on a convex function f: Rn → R:
From the definitions, it is clear that algorithms for some of the problems can be used to solve other problems in oracle-polynomial time:
The solvability of a problem crucially depends on the nature of K and the way K it is represented. For example:
Each of the above problems has a weak variant, in which the answer is given only approximately. To define the approximation, we define the following operations on convex sets:
Using these notions, the weak variants are:
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Algorithmic problems on convex sets
Many problems in mathematical programming can be formulated as problems on convex sets or convex bodies. Six kinds of problems are particularly important: optimization, violation, validity, separation, membership and emptiness. Each of these problems has a strong (exact) variant, and a weak (approximate) variant.
In all problem descriptions, K denotes a compact and convex set in Rn.
The strong variants of the problems are:
Closely related to the problems on convex sets is the following problem on a convex function f: Rn → R:
From the definitions, it is clear that algorithms for some of the problems can be used to solve other problems in oracle-polynomial time:
The solvability of a problem crucially depends on the nature of K and the way K it is represented. For example:
Each of the above problems has a weak variant, in which the answer is given only approximately. To define the approximation, we define the following operations on convex sets:
Using these notions, the weak variants are: